#7658163

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,16,20)(11,21,22,17)(12,23,14,18)(13,15,19,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,13)(14,19)(15,18)(16,22)(20,21)(23,24)(26,27)(31,33)(32,34)(38,39);(1,3,4,7)(2,5,6,8)(9,12,16,14)(10,18,20,23)(11,13,22,19)(15,21,24,17)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,13,16,19)(10,15,20,24)(11,12,22,14)(17,18,21,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,14,16,12)(10,18,20,23)(11,19,22,13)(15,21,24,17)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(9,15)(10,19)(11,18)(12,21)(13,20)(14,17)(16,24)(22,23)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565359);